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Mirrors > Home > ILE Home > Th. List > strle2g | GIF version |
Description: Make a structure from a pair. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.) |
Ref | Expression |
---|---|
strle1.i | ⊢ 𝐼 ∈ ℕ |
strle1.a | ⊢ 𝐴 = 𝐼 |
strle2.j | ⊢ 𝐼 < 𝐽 |
strle2.k | ⊢ 𝐽 ∈ ℕ |
strle2.b | ⊢ 𝐵 = 𝐽 |
Ref | Expression |
---|---|
strle2g | ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉} Struct 〈𝐼, 𝐽〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 3567 | . 2 ⊢ {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉} = ({〈𝐴, 𝑋〉} ∪ {〈𝐵, 𝑌〉}) | |
2 | strle1.i | . . . . 5 ⊢ 𝐼 ∈ ℕ | |
3 | strle1.a | . . . . 5 ⊢ 𝐴 = 𝐼 | |
4 | 2, 3 | strle1g 12280 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → {〈𝐴, 𝑋〉} Struct 〈𝐼, 𝐼〉) |
5 | 4 | adantr 274 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {〈𝐴, 𝑋〉} Struct 〈𝐼, 𝐼〉) |
6 | strle2.k | . . . . 5 ⊢ 𝐽 ∈ ℕ | |
7 | strle2.b | . . . . 5 ⊢ 𝐵 = 𝐽 | |
8 | 6, 7 | strle1g 12280 | . . . 4 ⊢ (𝑌 ∈ 𝑊 → {〈𝐵, 𝑌〉} Struct 〈𝐽, 𝐽〉) |
9 | 8 | adantl 275 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {〈𝐵, 𝑌〉} Struct 〈𝐽, 𝐽〉) |
10 | strle2.j | . . . 4 ⊢ 𝐼 < 𝐽 | |
11 | 10 | a1i 9 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → 𝐼 < 𝐽) |
12 | 5, 9, 11 | strleund 12278 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ({〈𝐴, 𝑋〉} ∪ {〈𝐵, 𝑌〉}) Struct 〈𝐼, 𝐽〉) |
13 | 1, 12 | eqbrtrid 3999 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉} Struct 〈𝐼, 𝐽〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1335 ∈ wcel 2128 ∪ cun 3100 {csn 3560 {cpr 3561 〈cop 3563 class class class wbr 3965 < clt 7912 ℕcn 8833 Struct cstr 12186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-addcom 7832 ax-addass 7834 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-0id 7840 ax-rnegex 7841 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-apti 7847 ax-pre-ltadd 7848 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-inn 8834 df-n0 9091 df-z 9168 df-uz 9440 df-fz 9913 df-struct 12192 |
This theorem is referenced by: strle3g 12282 2strstrg 12290 |
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